How to extrude outwards?
You can't really extrude outwards, it's actually a two step operation composed of an extrude and a scale commands.

Select the desired geometry and press E to extrude and immediately after Right Click to cancel the operation.

It will look like it id did nothing but it acualty created duplicate faces that are now overlapping the original.

Now just scale it outwards by pressing S.
You may optionaly press . (period key) before scaling to scale outwards relative to the 3D cursor position instead of the selection center.